We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell--Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. We obtain a combinatorially determined class of $K(\pi,1)$ spaces, and under a stronger combinatorial condition prove a factorization of the Poincaré polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk--Randell's formula relating the Poincaré polynomial to the lower central series of the fundamental group. This is joint work with Emanuele Delucchi.